CSCE222 First Exam

Lakukan tugas rumah & ujian kamu dengan baik sekarang menggunakan Quizwiz!

Existential Generalization (EG)

"Michelle got an A in the class." "Therefore, someone got an A in the class."

even integer

2k

Tuples

The ordered n-tuple (a1,a2,.....,an) is the ordered collection that has a1 as its first element and a2 as its second element and so on until an as its last element

pre-image

if the x for F(x)

Floor function

|_x_| is the largest integer less than or equal to x. 3.5=3

Implication (symbol)

Existential Instantiation (EI)

"There is someone who got an A in the course." "Let's call her a and say that a got an A"

De Morgan's Second Law

(A ∩ B)' = (A)' ∪ (B)'

De Morgan's First law

(A ∪ B)' = (A)' ∩ (B)'

Complementation law

(A=)=A *first equal sign denotes a double bar on top of A*

ordered pairs

2-tuples are called ordered pairs.

How many rows are there in a truth table with n propositional variables

2^(n)

odd integer

2k+1

if a set has n elements what is the cardinality of the *power set*

2ⁿ

Bijections

A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjective and injective). *basically a function as we usually know*

Geometric Progression

A geometric progression is a sequence of the form: where the initial term a and the common ratio r are real numbers.

Partial Functions

A partial function f from a set A to a set B is an assignment to each element a in a subset of A, called the domain of definition of f, of a unique element b in B ex: f: N → R where f(n) = √n is a partial function from Z to R where the domain of definition is the set of nonnegative integers. Note that f is undefined for negative integers.

Proper Subsets

A proper subset of a set A is a subset of A that is not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B. For example, if A={1,3,5} then B={1,5} is a proper subset of A. The set C={1,3,5} is a subset of A, but it is not a proper subset of A since C=A. The set D={1,4} is not even a subset of A, since 4 is not an element of A.

Sequences

A sequence is a function from a subset of the integers (usually either the set {0, 1, 2, 3, 4, .....} or {1, 2, 3, 4, ....} ) to a set S.

Complement laws

A ∪ A' = U A ∩ A' = ∅

Algorithms

An algorithm is a finite set of precise instructions for performing a computation or for solving a problem.

Proof by Contraposition

Assume ¬q and show ¬p is true also. This is sometimes called an indirect proof method. If we give a direct proof of ¬q → ¬p then we have a proof of p → q.

Distributive laws

A∩(B∪C)=(A∩B)∪(A∩C) A∪(B∩C)=(A∪B)∩(A∪C)

Associative laws

A∪(B∪C)=(A∪B)∪C A∩(B∩C)=(A∩B)∩C

Idempotent laws

A∪A=A A∩A=A

Commutative laws

A∪B=B∪A A∩B=B∩A

Domination laws

A∪U=U A∩∅=∅

Identity laws

A∪∅=A A∩U=A

2 things to know about subsets

Because a ∈ ∅ is always false, ∅ ⊆ S for every set S. Because a ∈ S → a ∈ S, S ⊆ S, for every set S.

Conjunction (Rules of Inference for Propositional )

Corresponding Tautology: ((p) ∧ (q)) →(p ∧ q) Example: Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math." "I will study English literature." "Therefore, I will study discrete math and I will study English literature."

Resolution

Corresponding Tautology: ((¬p ∨ r ) ∧ (p ∨ q)) →(q ∨ r) ¬p ∨ r p ∨ q ________ ∴q ∨ r

Simplification (Rules of Inference for Propositional )

Corresponding Tautology: (p∧q) →p Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math and English literature" "Therefore, I will study discrete math."

Modus Tollens

Corresponding Tautology: (¬q∧(p →q))→¬p Let p be "it is snowing." Let q be "I will study discrete math." "If it is snowing, then I will study discrete math." "I will not study discrete math." "Therefore , it is not snowing."

Hypothetical Syllogism

Corresponding Tautology: ((p →q) ∧ (q→r))→(p→ r) Let p be "it snows." Let q be "I will study discrete math." Let r be "I will get an A." "If it snows, then I will study discrete math." "If I study discrete math, I will get an A." "Therefore , If it snows, I will get an A."

Disjunctive Syllogism

Corresponding Tautology: (¬p∧(p ∨q))→q Let p be "I will study discrete math." Let q be "I will study English literature." "I will study discrete math or I will study English literature." "I will not study discrete math." "Therefore , I will study English literature."

Addition (Rules of Inference for Propositional )

Corresponding Tautology: p →(p ∨q) p _________________ ∴p∨q Let p be "I will study discrete math." Let q be "I will visit Las Vegas." "I will study discrete math." "Therefore, I will study discrete math or I will visit Las Vegas."

Arithmetic Progression

Definition: A arithmetic progression is a sequence of the form: where the initial term a and the common difference d are real numbers.

Fibonacci Sequence

Definition: Define the Fibonacci sequence, f0 ,f1 ,f2,..., by: Initial Conditions: f0 = 0, f1 = 1 Recurrence Relation: fn = fn-1 + fn-2

Factorial Function

Definition: f: N → Z+ , denoted by f(n) = n! is the product of the first n positive integers when n is a nonnegative integer. f(6) = 6! = 1 ∙ 2 ∙ 3∙ 4∙ 5 ∙ 6 = 720

Natural Numbers

Denoted by N {0,1,2,3....}

Integers

Denoted by Z {...,-3,-2,-1,0,1,2,3,...}

True or false? ∅ = { ∅ }

False ∅ ≠ { ∅ }

Inclusive Or

For p ∨q to be true, either one or both of p and q must be true

Truth Sets of Quantifiers

Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by {x ∈ D|P(x)} ex: The truth set of P(x) where the domain is the integers and P(x) is "|x| = 1" is the set {-1,1}

Complement

If A is a set, then the complement of the A (with respect to U), denoted by Ā is the set U - A Ā = {x ∈ U | x ∉ A} ex: If U is the positive integers less than 100, what is the complement of {x | x > 70} Solution: {x | x ≤ 70}

Implication

If p denotes "I am at home." and q denotes "It is raining." then p →q denotes "If I am at home then it is raining

Biconditional

If p denotes "I am at home." and q denotes "It is raining." then p ↔q denotes "I am at home if and only if it is raining."

Conjunction

If p denotes "I am at home." and q denotes "It is raining." then p ∧q denotes "I am at home and it is raining."

Disjunction

If p denotes "I am at home." and q denotes "It is raining." then p ∨q denotes "I am at home or it is raining."

Negation

If p denotes "The earth is round.", then ¬p denotes "It is not the case that the earth is round," or more simply "The earth is not round."

Cardinality

If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. |ø| = 0 Let S be the letters of the English alphabet. Then |S| = 26 |{1,2,3}| = 3 |{ø}| = 1

Functions

Let A and B be nonempty sets. A function f from A to B, denoted f: A → B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.

Difference

Let A and B be sets. The difference of A and B, denoted by A - B, is the set containing the elements of A that are not in B. The difference of A and B is also called the complement of B with respect to A. A - B = {x | x ∈ A x ∉ B} = A ∩B *A-B would leave only the A (without the intersection) shaded*

Union

Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set {x|x ∈ A V x ∈ B} ex: What is {1,2,3} ∪ {3, 4, 5}? Solution: {1,2,3,4,5} *the whole venn diagram*

Russell's Paradox

Let S be the set of all sets which are not members of themselves. A paradox results from trying to answer the question "Is S a member of itself?" ex Henry is a barber who shaves all people who do not shave themselves. A paradox results from trying to answer the question "Does Henry shave himself?"

Inverse Functions

Let f be a bijection from A to B. Then the inverse of f, denoted f^(-1), is the function from B to A defined as f^(-1)(y)=x if f(x)=y

Composition

Let f: B → C, g: A → B. The composition of f with g, denoted f∘g(g) is the function from A to C defined by f∘g(x)=f(g(x))

Universal Instantiation

Our domain consists of all dogs and Fido is a dog. "All dogs are cuddly." "Therefore, Fido is cuddly."

Universal Generalization (UG)

P(c) for an arbitrary c ------------------ ∴∀xP(x)

Example of Set builder notation

S = {x | x is a positive integer less than 100} Q+ = {x ∈ R | x = p/q, for some positive integers p,q}

{{1,2,3},a, {b,c}} {*N*,*Z*,*Q*,*R*} This is an example of...

Sets being elements of sets.

Modus Ponens

Tautology (p ∧ (p →q)) → q Let p be "It is snowing." Let q be "I will study discrete math." "If it is snowing, then I will study discrete math." "It is snowing." "Therefore , I will study discrete math."

Cartesian Product

The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A and b ∈ B ex: A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}

How is an empty set symbolized

The empty set is the set with no elements. Symbolized ∅, but {} also used

Intersection

The intersection of sets A and B, denoted by A ∩ B, is {x|x ∈ A ^ x ∈ B} ex: {1,2,3} ∩ {3,4,5} Solution: {3} {1,2,3} ∩ {4,5,6} ? Solution: ∅ *middle of venn diagram*

equality of ordered pairs

The ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.

Subsets

The set A is a subset of B, if and only if every element of A is also an element of B. The notation A ⊆ B is used to indicate that A is a subset of the set B.

Power Sets

The set of all subsets of a set A, denoted P(A), is called the power set of A. ex: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}

Symmetric Difference

The symmetric difference of A and B, denoted by A ⊕ B is the set (A-B) U (B-A) ex: U = {0,1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4,5} B ={4,5,6,7,8} What is A ⊕ B : Solution: {1,2,3,6,7,8} *everything but the middle of the venn diagram*

Exclusive Or

This is the meaning of Exclusive Or (Xor). In p ⊕ q , one of p and q must be true, but not both

Biconditional Statements

To prove a theorem that is a biconditional statement, that is, a statement of the form p ↔ q, we show that p → q and q →p are both true

Proof by Contradiction

To prove p, assume ¬p and derive a contradiction such as p ∧ ¬p. (an indirect form of proof). Since we have shown that ¬p →F is true , it follows that the contrapositive T→p also holds.

Showing that A is not a Subset of B

To show that A is not a subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.)

Showing that A is a Subset of B

To show that A ⊆ B, show that if x belongs to A, then x also belongs to B.

Set Equality

Two sets are equal if and only if they have the same elements. *no matter the order* {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5} are both equal sets

A tautology is...

a proposition which is always true

Absorption laws

a ∩ (a ∪ b)=a ∪ (a ∩ b)=a

Translate: You can access the Internet from campus only if you are a computer science major or you are not a freshman

a:You can access the internet from campus," c:"You are a computer science major," f: "You are a freshman." a→ (c ∨ ¬ f )

term of the sequence

a_n

codomain

are all the F(x)'s in F(x)=x

domain

are all the x's in F(x)=x

For interval notation name the symbol for *closed interval*

closed interval [a,b]

Asymmetric

if and only if (x,y) in R implies (y,x) not in R

Irreflexive

if and only if xRx does not hold for any x in A

reflexive

if and only if xRx for all x in A

Antisymmetric

if and only if xRy and yRx implies x = y

transitive

if and only if xRy and yRx implies xRz

symmetric

if and only xRy implies yRx

disjoint

if the intersection is empty, then A and B are said to be

image

is the *y*

Other names for functions

mappings or transformations.

For interval notation name the symbol for *open interval*

open interval (a,b)

one-to-one

or injective, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one. *passes the vertical line test*

onto

or surjective, if and only if for every element b∈B there is an element a∈A with f(a)=b . A function f is called a *surjection* if it is *onto.*

Translate: The automated reply cannot be sent when the file system is full

p: "The automated reply can be sent" q :"The file system is full." q→ ¬ p

converse of p →q and It raining is a sufficient condition for my not going to town

q →p If I do not go to town, then it is raining

universal set U

set containing everything currently under consideration

set of C

set of complex numbers.

set of Q

set of rational numbers

set of R

set of real numbers

Ceiling function

|-x-| is the smallest integer greater than or equal to x -1.5=-1

Negation (symbol)

¬

inverse of p → q and It raining is a sufficient condition for my not going to town

¬ p → ¬ q If it is not raining, then I will go to town.

contrapositive of p →q and It raining is a sufficient condition for my not going to town

¬q → ¬ p If I go to town, then it is not raining.

Biconditional (symbol)

Universal Quantifier

∀xP(x) - P(x) holds for all x in the domain

Uniqueness Quantifier

∃!x P(x) means that P(x) is true for one and only one x in the universe of discourse.

Some student in this class has visited Mexico. Let M(x) denote "x has visited Mexico" and S(x) denote "x is a student in this class," and U be all people.

∃x (S(x) ∧ M(x))

Every student in this class has visited Canada or Mexico Let M(x) denote "x has visited Mexico" and S(x) denote "x is a student in this class," and U be all people Add C(x) denoting "x has visited Canada

∃x (S(x)→ (M(x)∨C(x)))

Existential Quantifier

∃xP(x) - P(x) holds for some x in the domain D

Conjunction (symbol)

Disjunction (symbol)


Set pelajaran terkait

Ch 49 Management of Patients with Urinary Disorders

View Set

CH 42, Assessment and Concepts of Care for Patients with Eye and Vision Problems

View Set

Chapter: Completing the Application, Underwriting, and Delivering the Policy

View Set

Unité 8, p. 401 (Paris, capitale de la France)

View Set

LeeU Project Management Final Exam

View Set